dorsal/arxiv
View SchemaGeometric Phase, Curvature, and Extrapotentials in Constrained Quantum Systems
| Authors | Kevin A. Mitchell |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0001059 |
| URL | https://arxiv.org/abs/quant-ph/0001059 |
| DOI | 10.1103/PhysRevA.63.042112 |
Abstract
We derive an effective Hamiltonian for a quantum system constrained to a submanifold (the constraint manifold) of configuration space (the ambient space) by an infinite restoring force. We pay special attention to how this Hamiltonian depends on quantities which are external to the constraint manifold, such as the external curvature of the constraint manifold, the (Riemannian) curvature of the ambient space, and the constraining potential. In particular, we find the remarkable fact that the twisting of the constraining potential appears as a gauge potential in the constrained Hamiltonian. This gauge potential is an example of geometric phase, closely related to that originally discussed by Berry. The constrained Hamiltonian also contains an effective potential depending on the external curvature of the constraint manifold, the curvature of the ambient space, and the twisting of the constraining potential. The general nature of our analysis allows applications to a wide variety of problems, such as rigid molecules, the evolution of molecular systems along reaction paths, and quantum strip waveguides.
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"abstract": "We derive an effective Hamiltonian for a quantum system constrained to a\nsubmanifold (the constraint manifold) of configuration space (the ambient\nspace) by an infinite restoring force. We pay special attention to how this\nHamiltonian depends on quantities which are external to the constraint\nmanifold, such as the external curvature of the constraint manifold, the\n(Riemannian) curvature of the ambient space, and the constraining potential. In\nparticular, we find the remarkable fact that the twisting of the constraining\npotential appears as a gauge potential in the constrained Hamiltonian. This\ngauge potential is an example of geometric phase, closely related to that\noriginally discussed by Berry. The constrained Hamiltonian also contains an\neffective potential depending on the external curvature of the constraint\nmanifold, the curvature of the ambient space, and the twisting of the\nconstraining potential. The general nature of our analysis allows applications\nto a wide variety of problems, such as rigid molecules, the evolution of\nmolecular systems along reaction paths, and quantum strip waveguides.",
"arxiv_id": "quant-ph/0001059",
"authors": [
"Kevin A. Mitchell"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.63.042112",
"title": "Geometric Phase, Curvature, and Extrapotentials in Constrained Quantum Systems",
"url": "https://arxiv.org/abs/quant-ph/0001059"
},
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